knotkit

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knotkit is a C++ software package written by Cotton Seed ([email protected]) for computing some knot and manifold invariants appearing in low-dimensional topology. Other contributors include Josh Batson.

TABLE OF CONTENTS

1. INTRO

2. BUILDING

3. USAGE

4. UPCOMING CHANGES

5. FOR DEVELOPERS

6. INTRO

In addition to accepting knot presentations in a variety of formats (see usage below), knotkit contains the following tables of knot data:

• The Rolfsen knot tables through 10 crossings, extracted from Bar-Natan's Mathematica package, KnotTheory`,
• The Hoste-Weeks-Thistlewaite knot tables through 16 crossings taken from knotscape, and
• and the Morwen Thistlewaite hyperbolic link tables taken from SnapPy.

This version of knotkit has support to compute the following invariants:

• Khovanov homology See: M. Khovanov, A Categorification of the Jones Polynomial, Duke Math. J. 101 (2000), 359--426, arXiv:math/9908171. D. Bar-Natan, On Khovanov's Categorification of the Jones Polynomial, Algebraic and Geometric Topology 2 (2002), 337--370.

• Szabo's geometric spectral sequence Computing gss was the original motivation for writing knotkit. For more information on gss, see: Z. Szabo, A geometric spectral sequence in Khovanov homology, arXiv:1010.4252, and C. Seed, Computations of Szabo's Geometric Spectral Sequence in Khovanov Homology, arXiv:1110.0735.

• The Batson-Seed link splitting spectral sequence See: J. Batson, C. Seed, A Link Splitting Spectral Sequence in Khovanov Homology.

• The Lipshitz-Sarkar Steenrod square on Khovanov homology See: R. Lipshitz, S. Sarkar, A Khovanov Homotopy Type, arXiv:1112.3932. R. Lipshitz, S. Sarkar, A Steenrod Square on Khovanov Homology, arXiv:1204.5776.

• Bar-Natan's analogue of Lee homology See: E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), 2, 554–586, arXiv:math/0210213. D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443–1499, arXiv:math/0410495. R. Lipshitz, S. Sarkar, A refinement of Rasmussen's s-invariant, arXiv:1206.3532.

• The s-invariant coming from Bar-Natan's analogue of Lee homology In addition to the above references, see: J. Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 2 (2010), 419--447, arXiv:math/0402131. R. Lipshitz, S. Sarkar, A refinement of Rasmussen's s-invariant, arXiv:1206.3532.

2. BUILDING

To build knotkit on OS X, you will need the latest version of XCode (available for free in the App Store) and the GNU multiple precision arithmetic library (GMP). You can get GMP here:

http://gmplib.org/

knotkit also builds under Linux. In addition to GMP, you will need a C++ compiler which supports C++11. I use LLVM clang, but knotkit should build with a recent version of GCC.

knotkit doesn't have a sophisticated build system. To build knotkit, just run:

make

from the knotkit source directory. This should build the executable "kk". For instructions on invoking kk, see usage below. If you run into a problem, please contact me: Cotton Seed ([email protected]).

3. USAGE

The usage message for kk is given below. This can also be obtained by running "kk -h".

A note about output. The output for commands kh, gss, ls and lee is a .tex file which renders the bigraded homology group or spectral sequence. The output for the command sq2 matches the output for the program written by Lipshitz-Sarkar and is suitable for loading into Sage. The command s outputs a single line of text.

usage: kk [options...] compute for knot or link can be one of: kh: Khovanov homology gss: Szabo's geometric spectral sequence lsss: Batson-Seed link splitting spectral sequence component weights are 0, 1, ..., m sq2: Lipshitz-Sarkar Steenrod square on Z/2 Kh output suitable for Sage leess: spectral sequence coming from Bar-Natan analogue of Lee's deformation of Khovanov's complex (whew!) s: Rasmussen's s-invariant coming from lee output: kh, gss, lsss, leess: .tex file sq2: text in Sage format s: text options: -r : compute reduced theory -h : print this message -o : write output to (stdout is the default) -f : ground field (if applicable) (Z2 is the default) -v : verbose: report progress as the computation proceeds can be one of: Z2, Z3, Q

4. UPCOMING CHANGES

The following changes are currently planned:

• support for Z/p, p arbitrary prime and F(x) field of rational functions
• Roberts' totally twisted Khovanov homology
• the E^3 page of the twisted spectral sequence Kh(L) => \widehat{HF}(\Sigma_L)
• spectral sequences of PIDs: lsss over F[x], gss over Z
• maps induced by cobordisms

5. FOR DEVELOPERS

If you are interested in contributing to knotkit or using it for a new application, please contact me: Cotton Seed ([email protected]). I am interested in developing knotkit to be an open platform for performing computations arising in low-dimensional topology.